Equations Of Ellipse

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What is an Ellipse?

Ellipses for the first time was discovered by ancient greeks when they were studying the conic section. Ellipse was achieved when the right corner of it was sliced at different angles. Given below is a demonstration of how the right angle of an ellipse can be sliced at different angles.   

The resultant of the intersection of a right circular cone with a plane is called a conic section, or conic. Conic is basically a shape that is determined by the angle at which the plane intersects the cone. It can also be described by a set of points in the coordinate plane which can also be represented by the graph of any quadratic equation in two variables depending on the signs of the equations and the coefficients of the variable. 

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More About Ellipse

An ellipse and a circle look almost the same, the only difference is that an ellipse is slightly squashed into an oval. You can say it’s like a line bending around until the two of its ends meet, just like a circle. Things having a shape like an ellipse is known as 'elliptical'.

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Focus: Ellipse is denoted by a set of points(x,y) in a plane such that the sum of their distances from two fixed points is always constant. These fixed points are called foci (or focus if singular) of the ellipse.

Major Axis And Minor Axis: Every ellipse has two axes of symmetry called the major axis and minor axis. The longer axis known as the major axis has its endpoint as the vertex of the ellipse. Similarly, the shorter axis known as the minor axis has its endpoint as a co-vertex of the ellipse. The foci always rest on the major axis, and the sum of the distances between the foci and any other point on the ellipse (the constant sum) is always greater than the distance between the foci. 

Centre of an Ellipse: The centre of an ellipse is the common point and also the midpoint of both the major and minor axes. Both the axes (major and minor) are perpendicular to the centre. 


The General Equation of Ellipse

There is a standard form of the general equation of ellipse. 

\[\frac{x^2}{a^2}\] + \[\frac{y^2}{b^2}\] = 1

Ellipses are usually positioned in two ways - vertically and horizontally. Ellipses are said to be vertical in the coordinate plane if the axes on x– and y-axes whereas it is said to be horizontal in the coordinate plane if the axes lie parallel to the x– and y-axes. Apart from these, ellipses can also be rotated in the coordinate plane.

There are two cases to work with horizontal and vertical ellipses in the coordinate plane: 

i) Ellipses that are centred at the origin and 

ii) Ellipses that are centred at a point other than the origin. 

The general equation of ellipses in a standard form or say standard equation of ellipse is given below:

 \[\frac{x^2}{a^2}\] + \[\frac{y^2}{b^2}\]


Derivation of Equations of Ellipse

When the centre of the ellipse is at the origin and the foci are on the x-axis or y-axis, then the standard equation of ellipse can be derived as shown below.

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Now its time for deriving the standard equation of ellipse that is shown in Fig.5 (a) with the foci on the x-axis. Let F1 and F2 be the foci and O be the mid-point of the line F1F2. Also, let O be the origin and the line from O through F2 be the positive x-axis and that through F1 as the negative x-axis.

Further, let the line drawn through O perpendicular to the x-axis be the y-axis. Let the coordinates of F1 be (– c, 0) and F2 be (c, 0) as shown in Fig.5 (a) above.

Now, we take a point P(x, y) on the ellipse such that, PF1 + PF2 = 2a

By the distance formula, we have,

√{(x + c) 2 + y2} + √{(x – c) 2 + y2} = 2a

Or, √ {(x + c)2 + y2} = 2a – √ {(x – c)2 + y2}

Further, let’s square both sides. Hence, we have

(x + c) 2 + y2 = 4a2 – 4a√{(x – c) 2 + y2} + (x – c) 2 + y2

Simplifying the equation, we get √ {(x – c)2 + y2} = a – x(c/a)

Now we can square both the sides again and just simplify it further to get,

x2/a2 + y2/(a2 – c2) = 1

We know that c2 = a2 – b2. Therefore, we have x2/a2 + y2/b2 = 1

Therefore, we can say that any point on the ellipse satisfies the equation:

x2/a2 + y2/b2 = 1 …


Solved Examples

Example 1) Use the formula to find the coordinates of foci when the major axis is 5 and the minor axis is 3.

Solution 1) Given the formula   F = \[\sqrt{j^2-n^2}\]

F = \[\sqrt{5^2-3^2}\]

F = \[\sqrt{25-9}\]

F = \[\sqrt{16}\]

F = 4 

Foci = (0,4) & (0,-4)

Example 2) Use the formula to find the coordinates of foci when the major axis is 10 and the minor axis is 6.

Solution 2)  Given the formula   F = \[\sqrt{j^2-n^2}\]

F = \[\sqrt{10^2-6^2}\]

F = \[\sqrt{100-36}\]

F = \[\sqrt{64}\]

F = 8

Foci = (0,8) & (0,-8)

FAQ (Frequently Asked Questions)

Question 1: What is the eccentricity of an Ellipse?

Answer: Eccentricity = c/a. It basically tells us the measure of how much an ellipse can depart from a circle.

c = The distance between the centre and one of the two focuses (foci).

a = The distance between one of the focuses to the point at which the semi-minor axis meets the nearest perimeter or circumference. Just for the information, the semi-major axis is the distance between the centre and the farthest perimeter or circumference.

The focuses are also the centre of the circle. While A be the radius of the circle, c will be zero. So c/a would be equal to zero. A circle is nothing but a special case of an ellipse.

We can assume a long ellipse where a & c can be almost equal, therefore, c/a will approach unity.

Question 2: How to draw an Ellipse?

Answer: We all can draw an ellipse with the help of a piece of cardboard, two soft board pins, a pencil, and a string. First, we will have to attach the soft board pins on the cardboard as its foci, just like in an ellipse. Secondly, we will have to cut a length of string but we have to make sure that it is longer than the distance between the two soft board pins (the length of the string will represent the constant in the definition). Next, we just have to tack each end of the string to the cardboard and lastly outline a curve using a pencil firmly held against the string. The result will be an ellipse. We can draw an ellipse using a piece of cardboard, two thumbtacks, a pencil, and a string. Next, we have to place the thumbtacks on the cardboard to form the foci of the ellipse. Cut a piece of string longer than the distance between the two thumbtacks (the length of the string represents the constant in the definition). The final step is to tack each end of the string to the cardboard, and then trace a curve using a pencil held taut against the string. The result is an ellipse.